A priori estimation of scale and overall anisotropic temperature factors from the Patterson origin peak

An idea due to D. Rogers [Computing Methods in Crystallography (1965), edited by J. S. Rollett, pp. 117-148. Oxford: Pergamon Press] has been developed and implemented. The method is an advantageous alternative to Wilson plot or K-curve scaling of intensity data. On the relative experimental scale the structure factor can be written in matrix notation as F(h) = k^-1Σ_jf_j(h) exp (2πih^Tx_j) exp (-h_Tb_jh); and the squared structure-factor magnitude can be written as |F(h)|² = k^-2 exp (-2h^Tbh){Σ_jf²_j + 2Σ_j Σ_k>_jf_jf_k exp [2πih^T(x_j - x_k)]}, if a common, or average, anisotropic temperature factor is factored out of the atomic summations. The f_j² summation corresponds to the Patterson origin peak, and the f_j f_k double summation to the off-origin Patterson peaks. A trivariate Gaussian density function, P(u) - P_min = p₀ exp (-u^Tpu), is fitted by least squares to the origin peak from a Patterson synthesis with coefficients |F|²_meas/Σ_jf²_j. Fourier inversion of the fitted Gaussian gives the scale and thermal parameters, k² = (det p)^1/2/(π^3/2V_cellP₀) and b = (π²/2)p^-1. The fit of the parameter P_min is constrained by the condition that P_min = -F(000)²/(k² V_cellΣ_jZ²_j), and thus only P₀ and the six coefficients P_ij (i < j = 1, 2, 3) are independent parameters.